The supercritical birth, death and catastrophe process: Limit theorems on the set of non-extinction

Abstract

The skip-free property of the super-critical linear birth and death process with a linear catastrophe component is exploited to obtain an almost sure convergence theorem for the population size without any extra moment assumptions. This completes some earlier work of the author. A proof is sketched of a central limit theorem for the logarithm of the population size. Identification of the form of the norming constants for this result requires estimates of the extinction probabilities as functions of the initial population size. This is related to moment conditions on the decrement distribution.